nLab
corepresentable functor

A functorC→SetC \to Set is corepresentable if it is (isomorphic to) a functor of the form HomC(c,−)Hom_C(c,-) for some objectc∈Cc\in C.

This is equivalently a representable functor defined on the opposite categoryCopC^{op}. Often no terminological distinction is made between representable and corepresentable ones (both being called simply “representable”), since a functor C→SetC\to Set can only be “corepresentable” while a functor Cop→SetC^{op}\to Set can only be “representable”.

Particularly in the study of moduli problems, there is also the notion of corepresentable contravariant functors. A functor F:Cop→SetF : C^op \to Set is corepresentable if and only if there exists an object XX in CC and a morphism F→hYF \to h_Y such that for any object TT in CC, the canonical map HomC(X,T)→HomPSh(C)(hX,hT)→HomPSh(C)(F,hT)Hom_C(X,T) \to Hom_{\PSh(C)}(h_X, h_T) \to Hom_{PSh(C)}(F, h_T) is bijective.

Revised on June 5, 2017 17:47:07
by Ingo Blechschmidt
(2a02:810d:e80:3514:b533:497:dd42:eb33)